Question

If $\cos \frac{A}{2}=\sqrt{\frac{b+c}{2c}}$,then prove that $a^2+b^2=c^2$

Answer 1

Given, $\cos \frac{A}{2}=\sqrt{\frac{b+c}{2c}}$
$\Rightarrow \frac{s(s-a)}{bc}=\frac{b+c}{2c}$
$2s(2s-2a)=2b(b+c)$
$(b+c+a)(b+c-a)=2b^2+2bc$
$(b+c{)}^2-a^2=2b^2+2bc$ $\therefore c^2=a^2+b^2$